On Makarov's principle in conformal mappingON MAKAROV’S PRINCIPLE IN CONFORMAL MAPPING 3 1.3....

ON MAKAROV’S PRINCIPLE IN CONFORMAL MAPPING

OLEG IVRII

Abstract. We examine several characteristics of conformal maps thatresemble the variance of a Gaussian: asymptotic variance, the constantin Makarov’s law of iterated logarithm and the second derivative of theintegral means spectrum at the origin. While these quantities neednot be equal in general, they agree for domains whose boundaries areregular fractals such as Julia sets or limit sets of quasi-Fuchsian groups.We give a new proof of these dynamical equalities. We also show thatthese characteristics have the same universal bounds and prove a centrallimit theorem for extremals. Our method is based on analyzing the localvariance of dyadic martingales associated to Bloch functions.

1. Introduction

Let Ω ⊂ C be a simply connected domain in the plane whose boundary

is a Jordan curve. In 1985, N. Makarov [18] introduced probabilistic tech-

niques into the theory of conformal mapping to show that the harmonic

measure on ∂Ω necessarily has Hausdorff dimension 1. This is quite sur-

prising for domains Ω with H.dim ∂Ω > 1: for such domains, Makarov’s

theorem suggests that Brownian motion started at an interior point z0 ∈ Ω

only hits a small subset of the boundary. Later, Makarov obtained finer

estimates for the metric properties of harmonic measure by proving a law of

iterated logarithm for Bloch functions. Central to Makarov’s reasoning was

the following insight: Bloch functions behave like martingales.

1.1. Three characteristics of Bloch functions. By definition, the Bloch

space consists of holomorphic functions on the unit disk for which the Bloch

norm ‖b‖B := supz∈D |b′(z)|(1−|z|2) is finite. Bloch functions naturally arise

from conformal mappings: if f is a conformal mapping, then ‖ log f ′‖B 6 6.

While a Bloch function b ∈ B can grow like |b(z)| ∼ log 11−|z| , the martingale

property of b suggests that the average rate of growth is much smaller –

2010 Mathematics Subject Classification. Primary 30C35; Secondary 30C62.The author was supported by the Academy of Finland, project nos. 271983 and 273458.

1

2 O. IVRII

approximately the square root of the maximum. With this in mind, it is

natural to expect that the following three characteristics are finite:

• The asymptotic variance

σ2(b) = lim supr→1

1

2π| log(1− r)|

ˆ|z|=r

|b(z)|2 |dz|. (1.1)

• The LIL constant

CLIL(b) = ess supθ∈[0,2π)

lim supr→1

|b(reiθ)|√log 1

1−r log log log 11−r

. (1.2)

• The integral means spectrum

βb(τ) = lim supr→1

1

| log(1− r)|· log

ˆ|z|=r

∣∣eτb(z)∣∣ |dz|, τ ∈ C. (1.3)

The fact “C2LIL(b) 6 C1‖b‖2B” is Makarov’s celebrated law of iterated log-

arithm mentioned above. The inequalities σ2(b) 6 C2‖b‖2B and βb(τ) 6

C3|τ |2‖b‖2B are also well known and can be found in Pommerenke’s book

[24]. We will reprove these facts here with an emphasis on sharp constants.

1.2. Dynamical equalities. In dynamical situations, the above character-

istics are linked by an explicit relation:

Theorem 1.1. Suppose f(z) is a conformal map, such that the image of the

unit circle f(S1) is a Jordan curve, invariant under a hyperbolic conformal

dynamical system. Then,

2d2

dτ2

∣∣∣∣τ=0

βlog f ′(τ) = σ2(log f ′) = C2LIL(log f ′). (1.4)

The equalities in (1.4) are mediated by a fourth quantity involving the

dynamical asymptotic variance of a Holder continuous potential from ther-

modynamic formalism. Theorem 1.1 has a rich history: the connection with

C2LIL is due to Przytycki, Urbanski, Zdunik [25], with integral means due to

Makarov and Binder [21, 5], and with σ2 by McMullen [22]. See also [1] for

additional details. One of our central objectives is to give a new proof of

Theorem 1.1 that does not involve thermodynamic formalism. Instead, we

work with a new central quantity: the local variance of a dyadic martingale

associated to a Bloch function. The definition will be given in Section 2.

We emphasize that the above quantities are unrelated for general Bloch

functions. We refer the reader to [3, 17] for interesting examples.

ON MAKAROV’S PRINCIPLE IN CONFORMAL MAPPING 3

1.3. Universal bounds. It is natural to ask if the above characteristics

agree on the level of universal bounds, taken over all conformal maps. We

show that this is essentially the case; however, in order to be able to localize

these characteristics, we are forced to restrict to conformal maps that have

quasiconformal extensions with bounded distortion.

To be concrete, let S be the class of conformal maps f : D→ C normalized

so that f(0) = 0 and f ′(0) = 1, and for 0 < k < 1, let Sk ⊂ S denote the

collection of maps that admit a k-quasiconformal extension to the complex

plane. Let Bk(τ) := supf∈Sk βlog f ′(τ).

Theorem 1.2. For any 0 < k < 1,

limτ→0

Bk(τ)

|τ |2/4= sup

f∈Skσ2(log f ′) = sup

f∈SkC2

LIL(log f ′).

In fact, we will show that for each of the three expressions, it suffices

to take the supremum over fractal objects that satisfy Theorem 1.1. The

above quantity will be denoted Σ2(k). As discussed in [13], Σ2(k)/k2 is a

non-decreasing convex function of k. It is currently known that

0.93 < limk→1−

Σ2(k) < (1.24)2.

We refer the reader to [1, Section 8] for the lower bound and to [10, 11, 12]

for the upper bound. A theorem of Makarov [19], [7, Theorem VIII.2.1]

shows:

Corollary 1.3. (i) Let Ω = f(D) be the image of the unit disk and z0 be

a point in Ω. The harmonic measure ωz0 on ∂Ω, as viewed from z0, is

absolutely continuous with respect to the Hausdorff measure Λh(t),

h(t) = t exp

C

√log

1

tlog log log

1

t

, 0 < t < 10−7,

for any C > CLIL(bf ). In particular, C =√

Σ2(k) works.

(ii) Conversely, if C <√

Σ2(k), there exists a conformal map f ∈ Sk for

which ωz0 ⊥ Λh(t).

The connections to LIL in Theorem 1.2 and Corollary 1.3 were originally

proved together with I. Kayumov using a different method than presented

here.

4 O. IVRII

In Section 6, we will show that extremal Bloch functions obey a central

limit theorem. For a fixed r < 1, we may consider

br(θ) :=b(reiθ)√| log(1− r)|

(1.5)

as a random variable with respect to the probability measure |dz|/2π.

Theorem 1.4. Let 0 < k < 1 be fixed. Suppose f ∈ Sk and b = log f ′.

Given ε > 0, there exists a δ > 0 such that if r is sufficiently close to 1 and

σ2(b, r) =1

2π| log(1− r)|

ˆ|z|=r

|b(z)|2 |dz| > Σ2(k)− δ,

then the distribution of br is close to a complex Gaussian of mean 0 and

variance Σ2(k), up to an additive error of at most ε. In other words, Re br

and Im br approximate independent real Gaussians of variance Σ2(k)/2.

Loosely speaking, extremality forces fractal structure.

1.4. Infinitesimal analogue. Suppose µ ∈M(D) is a Beltrami coefficient

on the unit disk with ‖µ‖∞ 6 1. For t ∈ D, let wtµ be the principal solution

to the Beltrami equation ∂w = tµ ∂w. By construction, ϕt = wtµ∣∣D∗ is a

holomorphic family of conformal mappings of the exterior unit disk, with

ϕ0 being the identity map. In this setting, t → logϕ′t is a Banach-valued

holomorphic function (to the Bloch space of the exterior unit disk) whose

derivative at the origin is the Beurling transform

Sµ(z) = − 1

π

ˆD

µ(w)

(z − w)2|dw|2, |z| > 1, (1.6)

i.e.

‖ log(wtµ)′ − tSµ‖B(D∗) = O(|t|2), t ∈ D. (1.7)

We refer the reader to [13, Section 2] for details. Consider the rescaled

integral means spectrum

B∗(τ) := limk→0

Bk(τ/k) = sup|µ|6χD

βSµ(τ). (1.8)

Since the collection of Bloch functions Sµ, |µ| 6 χD is invariant under

rotation by eiθ ∈ S1, B∗(τ) only depends on |τ |. Whether or not Bk(τ) is

radially-invariant is an open problem. We have:


Corollary 1.5.

limτ→0

B∗(τ)

|τ |2/4= sup|µ|6χD

σ2(Sµ) = sup|µ|6χD

C2LIL(Sµ).

We denote the common quantity by Σ2. From the above construction,

it is clear that Σ2 = limk→0 Σ2(k)/k2. The quantity Σ2 was first studied in

[1], where it was established that 0.87913 6 Σ2 6 1, while Hedenmalm [9]

proved the strict inequality Σ2 < 1.

1.5. Notes and references. In [1], the original motivation for investigat-

ing Σ2 arose from the question of finding D(k), the maximal Minkowski

dimension of a k-quasicircle. The anti-symmetrization procedure of [16, 26]

and the relation [24, Corollary 10.18]

βf (t) = t− 1 ⇐⇒ t = M.dim f(S1), f ∈ Sk, (1.9)

reduce the problem of finding D(k) to investigating the function Bk(τ). In

[13], the author modified the argument of Becker and Pommerenke [4] for

estimating integral means to show the asymptotic expansion

D(k) = 1 + Σ2k2 +O(k8/3−ε). (1.10)

Together with Hedenmalm’s estimate Σ2 < 1, this improves on Smirnov’s

bound D(k) 6 1 + k2 from [26]. In this paper, we use Makarov’s Bloch

martingale technique to give an alternative perspective on the results from

[13]. In Section 5, we give an estimate for B∗(τ) which implies (1.10), albeit

with a slightly weaker error term.

An a priori difficulty in studying Σ2 is that the extremal problem of max-

imizing asymptotic variance sup|µ|6χD σ2(Sµ) has infinitely many solutions.

For instance, one can take an extremal µ and modify it in an arbitrary man-

ner on a compact subset of the disk. Alternatively, one can pullback an

extremal µ(z)dzdz by a Blaschke product B : D → D. Further, given two ex-

tremals µ, ν, one can glue them together µ · χRe z<0+ν · χRe z>0 to form

yet another extremal. These facts follow from the localization principle, see

[13, Section 4].

In [1, Section 6], extremal Bloch functions were studied indirectly via

fractal approximation. The idea was to show

Σ2 = supµ∈MI, |µ|6χD

σ2(Sµ), (1.11)

6 O. IVRII

where MI is the class of Beltrami coefficients that are eventually-invariant

under z → zd for some d > 2, i.e. satisfying (zd)∗µ = µ in some open

neighbourhood of the unit circle. There, the fractal approximation technique

was used to deduce Σ2 6 1 from Smirnov’s 1 + k2 bound.

Surprisingly, fractal approximation fails in the Fuchsian setting:

Σ2F = sup

µ∈MF, |µ|6χD

σ2(Sµ) < 2/3 < Σ2, (1.12)

where MF is the class of Beltrami coefficients invariant under some co-

compact Fuchsian group. By the work of McMullen [22], the problem of

understanding Σ2F is naturally related to maximizing the quotient of the

Weil-Petersson and Teichmuller metrics

Σ2F =

2

3· sup

(g,X,q)

(‖q‖T‖q‖WP

)2

,

where the supremum is taken over all genera g > 2, Riemann surfacesX ∈ Tgand quadratic differentials q ∈ Q(X). Here, we normalize the hyperbolic

area of a Riemann surface to be 1 so that the above quotient is invariant

under taking finite covers. See [1, Section 7] for details.

In contrast, Theorem 1.4 can be viewed as an attempt to study extremals

directly. Inspired by an analogy with a problem in superconductivity in

the Euclidean setting, Hedenmalm [9] suggested that extremals satisfy much

stronger statistical properties. Understanding the fine structure of extremals

would be a major advance in the field. For a possible approach, see the recent

work of A. Wennman [29].

In a companion paper [14], joint with I. Kayumov, we show that the

equality of universal characteristics remains valid if one takes the supremum

over the Bloch unit ball:

Σ2B := sup

‖b‖B61σ2(b) = sup

‖b‖B61C2

LIL(b) = limτ→0

1

|τ |2/4· sup‖b‖B61

βb(τ).

In addition, we also give the estimate Σ2B < min(0.9,Σ2).

Notation. Let ρ∗(z) = 2|z|2−1

be the density of the hyperbolic metric on the

exterior unit disk D∗ and ρH(z) = 1/y be the corresponding density in the

upper half-plane. To compare quantities, we use A & B to denote A > C ·Bfor some constant C > 0. The notation

fflf(t) g(t)dt denotes the average

value of the function f(t) with respect to the measure g(t)dt.


2. Background in probability

In this section, we discuss martingale analogues of the characteristics of

conformal maps mentioned in the introduction. We show that they are

controlled by the local variation.

2.1. Martingales and square functions. Let p > 2 be an integer, Dk be

the collection of p-adic intervals[j · p−k, (j + 1) · p−k

]contained in [0, 1],

and Mk be the σ-algebra generated by Dk. A (complex-valued) p-adic

martingale X on [0, 1] is a sequence of functions Xk∞k=0 such that

(i) Xk is measurable with respect to Mk,

(ii) E(Xk|Mk−1) = Xk−1.

We typically view X as a function from⋃∞k=0Dk to the complex numbers

which satisfies the averaging property

XI =1

p

p∑i=1

XIi , (2.1)

where the sum ranges over the p-adic children of I. For a point x ∈ [0, 1],

let I〈k〉(x) ∈ Dk denote the p-adic interval of length p−k containing x, and

∆〈k〉(x) = XI〈k〉(x) − XI〈k−1〉(x) be the jump at step k. The p-adic square

function is given by

〈X〉n :=n∑k=1

|∆〈k〉(x)|2. (2.2)

We say that a martingale has bounded increments if |∆〈k〉(x)| < C for all

x ∈ [0, 1] and k > 1. For such martingales, we consider:

• The asymptotic variance

σ2(X) = lim supn→∞

1

n

ˆ 1

0|Xn(x)|2 dx

= lim supn→∞

1

n

ˆ 1

0〈X〉n dx.

(The equality follows from the orthogonality of the jumps.)

• The LIL constant

CLIL(X) = ess supx∈[0,1]

lim supn→∞

|Xn(x)|√n log logn

.

• The integral means spectrum

βX(τ) = lim supn→∞

1

n· log

ˆ 1

0

∣∣eτXn(x)∣∣ dx, τ ∈ C.

8 O. IVRII

2.2. Local variance. For a p-adic interval I, we define the local variance

of X at I as

VarI X = Varp,I X =1

p

p∑i=1

|XIi −XI |2. (2.3)

More generally, we can consider

VarnI X =1

n

[1

pn

pn∑i=1

|XIi −XI |2], (2.4)

where we sum over all p-adic grandchildren of I of length p−n|I|. Polarizing,

we obtain the notion of local covariance

VarnI (X,Y ) =1

n

[1

pn

pn∑i=1

(XIi −XI)(YIi − YI)]

(2.5)

of two p-adic martingales X and Y . Our aim is to show that the local

variance controls the above characteristics:

Theorem 2.1. Suppose S is a real-valued martingale with bounded incre-

ments. Let m = infI VarI S and M = supI VarI S. Then,

(i) For a.e. x ∈ [0, 1],

m 6 lim inf〈S〉nn6 lim sup

〈S〉nn6 M,

(ii) m 6 σ2(S) 6M ,

(iii) m 6 (1/2) · C2LIL(S) 6M ,

(iv) For t ∈ R,

m 6 lim inft→0

βS(t)

t2/26 lim sup

t→0

βS(t)

t2/26M.

To evaluate the LIL constant of a martingale, we use a result of W. Stout

[27], which is stated explicitly in the form below in [20, Theorem 2.6]:

Lemma 2.2 (Stout). If Sn is a real-valued martingale with bounded incre-

ments, then

lim supn→∞

|Sn(x)|√2〈S〉n log log〈S〉n

= 1, (2.6)

almost surely on the set x : 〈S〉∞ =∞ where 〈S〉∞ = limn→∞〈S〉n.

Proof of Theorem 2.1. (i) Consider the auxiliary martingale T with T[0,1] =

0 and jumps

TIi − TI := |SIi − SI |2 −1

p

p∑j=1

|SIj − SI |2,


where I1, I2, . . . , Ip are the p-adic children of I. Note that T has bounded

increments since S does. Applying Lemma 2.2 to the martingale T , we see

that

Tn(x) = O(√

n log log n)

= o(n), for a.e. x ∈ [0, 1].

In particular,

〈S〉nn

=1

n

n∑k=1

VarI〈k〉(x) S + o(1), for a.e. x ∈ [0, 1]. (2.7)

The rest is easy: (ii) is trivial, (iii) follows from (i) by Stout’s lemma, while

(iv) follows from the expansion

1

p

p∑j=1

exp(t(SIj − SI)

)= 1 +

t2

2

(1

p

p∑j=1

|SIj − SI |2)

+O(t3).

This proof is complete.

The same argument shows:

Lemma 2.3. If two real-valued martingales S1, S2 satisfy VarI S1 = VarI S2

for all I, then they have the same LIL constant. More generally,

(1/2) |C2LIL(S1)− C2

LIL(S2)| 6 supI|VarI S1 −VarI S2|.

The above lemma also holds for the other characteristics discussed in

Theorem 2.1.

2.3. Some useful facts. For future reference, we record two martingale

estimates. Assume for simplicity that S is a real-valued dyadic martingale

with S[0,1] = 0 and |∆〈k〉(x)| 6 1. The sub-Gaussian estimate says that∣∣x ∈ [0, 1] : |Sn| > t∣∣ 6 e−ct2/n (2.8)

for some c > 0. The sub-Gaussian estimate is a consequence of a more

general statement, see [20, Proposition 2.7]. Another proof is given in [8].

Integrating (2.8), we obtain bounds for the moments

1

Γ(p+ 1)

ˆ 1

0|Sn|2p dx 6 (Cn)p, p > 0. (2.9)

10 O. IVRII

3. Bloch martingales

In this section, we review Makarov’s construction which associates a

dyadic martingale to a Bloch function. For convenience, we work in the

upper half-plane where the computations are slightly simpler. Therefore,

let us imagine that b is a holomorphic function on H with

‖b‖B(H) = supz∈H

2y · |b′(z)| 6 1. (3.1)

Here, we assume that b lies in the Bloch unit ball in order to not have to

write the Bloch norm all the time.

A dyadic interval I = [x1, x2] ⊂ [0, 1] defines a 1-box

I =w : Rew ∈ [x1, x2], Imw ∈ [(x2 − x1)/2, x2 − x1]

in the upper half-plane. The n-box nI is defined as the union of 1-boxes

associated to I and to all dyadic intervals contained in I of length at least

2−n+1|I|. For instance,

n[0,1] =w : Rew ∈ [0, 1], Imw ∈ [2−n, 1]

.

We use zI = (x1 + x2)/2 + (x2 − x1)i to denote the midpoint of the top

edge of I . Following Makarov [20], to the Bloch function b we associate

the complex-valued dyadic martingale B given by

BI = limy→0+

1

|I|

ˆIb(x+ iy)dx. (3.2)

Makarov showed that the above limit exists and satisfies

|b(zI)−BI | = O(1). (3.3)

In particular,

|BI −BJ | 6 C, (3.4)

whenever I, J are adjacent dyadic intervals of the same size. This is stronger

than simply saying that B has bounded increments because I, J may have

different parents. Makarov [20] observed that the converse also holds: if a

dyadic martingale on [0, 1] satisfies the above property, it comes from some

Bloch function b(z). It is therefore natural to refer to martingales satisfying

(3.4) as Bloch martingales.

One may view dyadic martingales abstractly, defined on the dyadic tree.

The notion of a Bloch martingale, however, requires an identification of the

dyadic tree with [0, 1]. One useful fact to keep in mind is:


Lemma 3.1 (Transmutation principle). For an abstract dyadic martingale

with bounded increments, there is an embedding to [0, 1] so that it is Bloch.

Sketch of proof. Let X be a dyadic martingale on [0, 1] with increments

|∆〈k〉(x)| 6 C. We claim that by flipping the branches at a countable set of

dyadic intervals, we can turn X in a dyadic martingale satisfying (3.4) with

the same constant C. The proof proceeds by induction. Suppose the jump

discontinuities of Xk (viewed as a function on [0, 1]) at the dyadic points

j/2k, j = 0, 1, 2, . . . , 2k are bounded by C. Let us examine Xk+1. At the

“new” dyadic points j/2k+1 with j odd, the jumps are bounded by C by

the assumption on the increments, so one has to only worry about the old

dyadic points. It is easy to see that by flipping dyadic intervals of length

1/2k+1, one can make sure that these jumps do not stack up.

The transmutation principle allows one to prove general estimates for

dyadic martingales such as (2.9) by working analytically in the realm of

Bloch functions.

We define the asymptotic variance, LIL constant and integral means of a

Bloch function b ∈ B(H) as

1

log 2· σ2(B),

1

log 2· C2

LIL(B),1

log 2· βB(t),

respectively. The factor χ = log 2 comes from the height of the boxes in

the dyadic grid (as measured in the hyperbolic metric). It plays the role

of the Lyapunov exponent, cf. [22, Theorem 2.7]. If one instead works

with the p-adic grid, then the normalizing factor would be χ = log p. As a

consequence of Theorem 3.2 below, the definition of the asymptotic variance

can be expressed in function-theoretic terms as

σ2[0,1](b) = lim sup

y→0+

1

| log y|

ˆ 1

0|b(x+ iy)|2dx, (3.5)

= lim suph→0+

1

| log h|

ˆ 1

h

ˆ 1

0

∣∣∣∣2b′(x+ iy)

ρH

∣∣∣∣2 |dz|2y . (3.6)

More generally, in [22, Section 6], McMullen showed that one can compute

the asymptotic variance by examining Cesaro averages of integral means

that involve higher order derivatives.

12 O. IVRII

3.1. A formula for the local variance.

Theorem 3.2. Suppose I ⊂ [0, 1] is a dyadic interval and I1, I2, . . . , I2n are

its dyadic grandchildren of length 2−n|I|. Then,

1

log 2·Varn2,I B =

ˆbot(nI )

|b(z)−BI |2dx+O(‖b‖2B/

√n)

(3.7)

=

nI

∣∣∣∣2b′ρH (z)

∣∣∣∣2 |dz|2y +O(‖b‖2B/

√n), (3.8)

where in (3.7), we integrate over the bottom side of nI . Furthermore, we

have the “complexification” relations

Varn2,I(ReB) =1

2·Varn2,I B +O

(‖b‖2B/

√n)

(3.9)

and

Varn2,I(ReB, ImB) = O(‖b‖2B/

√n). (3.10)

Proof. Due to scale invariance, we only need to consider the case when

I = [0, 1] and ‖b‖B(H) = 1. We first make some preliminary observations.

Since B has bounded jumps,

〈B〉n . n and1

2n

2n∑j=1

|BIj −BI |2 . n.

The Cauchy-Schwarz inequality gives

1

2n

2n∑j=1

|BIj −BI | .√n. (3.11)

The Bloch property impliesˆbot(nI )

|b(z)− b(zI)| ds .√n (3.12)

and ∣∣∣∣ˆbot(nI )

|b(z)− b(zI)|2 ds−1

2n

2n∑j=1

|BIj −BI |2∣∣∣∣ . √n. (3.13)

Following [23], we apply Green’s identityˆΩ

(u∆v − v∆u)dxdy =

ˆ∂Ω

(u · ∂nv − v · ∂nu)ds,

where ∂n refers to differentiation with respect to the outward unit normal

vector and ds denotes integration with respect to arc length. The choice

Ω = nI , u = y, v = |b(z)− b(zI)|2


yields ˆnI

y · |2b′(z)|2 dxdy =

ˆnI

y ·∆|b(z)− b(zI)|2 dxdy

= −ˆ∂nI

∂ny · |b(z)− bI(z)|2 ds+

ˆ∂nI

y · ∂n|b(z)− b(zI)|2 ds.

It is evident that

−ˆ∂nI

∂ny · |b(z)− b(zI)|2 ds =

ˆbot(nI )

|b(z)− b(zI)|2 ds−O(1). (3.14)

For the error term,ˆ∂nI

y · ∂n|b(z)− b(zI)|2 ds .ˆ∂nI

|b(z)− b(zI)| ds.

From the definition of a Bloch function (3.1), the integral of |b(z) − b(zI)|over the top, left and right sides of ∂nI is O(1), while according to (3.12),

the integral over the bottom side is O(√n). Summarizing, we see that

nI

∣∣∣∣2b′(z)ρH

∣∣∣∣2 |dz|2y =1

n log 2

ˆbot(nI )

|b(z)− b(zI)|2 ds+O(1/√n). (3.15)

Combining with (3.13) gives another error of O(1/√n) and proves (3.8).

Repeating the above argument with v = |Re b− Re b(zI)|2 gives

Varn2,I(ReB)

log 2=

ˆbot(nI )

|Re b(z)− ReBI |2dx+O(‖b‖2B/

√n)

(3.16)

=1

2

nI

∣∣∣∣2∇(Re b)

ρH(z)

∣∣∣∣2 |dz|2y +O(‖b‖2B/

√n), (3.17)

which proves (3.9) since |b′| = |∇(Re b)|. Finally (3.10) follows from polar-

ization and the identity ∇(Re b) · ∇(Im b) = 0 which is a consequence of the

Cauchy-Riemann relations.

3.2. Complexification.

Theorem 3.3 (Complexification). For any Bloch function b ∈ B(H),

σ2(Re b) = (1/2) · σ2(b), CLIL(Re b) = CLIL(b).

Proof. The first statement is an immediate consequence of (3.9). For the

second statement, note that the function θ → CLIL(Re eiθB) is continuous

and

CLIL(B) = supθ∈[0,2π)

CLIL(Re eiθB).

14 O. IVRII

We must therefore show CLIL(Re eiθB) 6 CLIL(ReB), for any θ ∈ [0, 2π).

However, if view ReB and Re eiθB as p-adic martingales with p = 2n large,

then by (3.9), their local variances are approximately equal. The assertion

now follows from Lemma 2.3.

Remark. For lacunary series, the equality CLIL(Re b) = CLIL(b) goes back

to the 1959 work of M. Weiss [28].

4. The Box Lemma

Let Hk denote the class of conformal maps f : H → C which admit

a k-quasiconformal extension to the plane and fix the points 0, 1,∞. For

f ∈ Hk, we denote the associated Bloch function by bf = log f ′ ∈ B(H). It

is not difficult to show that the expressions

supf∈Hk

σ2(bf ), supf∈Hk

C2LIL(bf ), sup

f∈Hk

βbf (t),

coincide with their analogues for the class Sk from the introduction. The

proof of Theorem 1.2 is now completed by the Box Lemma from [13] which

describes the average non-linearity nf := f ′′/f ′ = (log f ′)′ of conformal

mappings:

Lemma 4.1. (i) Fix 0 < k < 1. Given ε > 0, there exists n > 1 sufficiently

large so that for any n-box nI ⊂ H and any conformal map f ∈ Hk,

nI

∣∣∣∣2nfρH (z)

∣∣∣∣2 |dz|2y < Σ2(k) + ε. (4.1)

(ii) Conversely, for any ε > 0, there exists a conformal map f ∈ Hk,

whose dilatation dil. f := ∂f/∂f is periodic with respect to the 2n-adic grid

for some n > 1, and which satisfies nI

∣∣∣∣2nfρH (z)

∣∣∣∣2 |dz|2y > Σ2(k)− ε, (4.2)

on every n-box nI .

Remark. The proof given in [13] forces us to restrict our attention to classes

of conformal maps with bounded distortion. It would be interesting to

know if a variant of the box lemma holds for all conformal maps with

limk→1− Σ2(k) in place of Σ2(k).


In view of Theorems 2.1 and 3.2, (i) gives the upper bound in Theorem 1.2,

while (ii) gives the lower bound. The notion of periodic Beltrami coefficients

will be discussed below in Section 4.1. In order to state the infinitesimal

version of the box lemma, note that the formula for the Beurling transform

(1.6) may not converge if µ is not compactly supported. Therefore, we are

obliged to work with a modified Beurling transform

S#µ(z) = − 1

π

ˆHµ(ζ)

[1

(ζ − z)2− 1

ζ(ζ − 1)

]|dζ|2. (4.3)

However, the formula for the derivative remains the same:

“(Sµ)′(z)” := (S#µ)′(z) = − 2

π

ˆH

µ(ζ)

(ζ − z)3|dζ|2. (4.4)

In [13], the infinitesimal analogue of the box lemma was proved with a

quantitative relation between the box size and the error term:

Lemma 4.2. (i) For any Beltrami coefficient µ with |µ| 6 χH and n-box

nI ⊂ H, nI

∣∣∣∣2(Sµ)′

ρH(z)

∣∣∣∣2 |dz|2y < Σ2 + C/n. (4.5)

(ii) Conversely, for n > 1, there exists a Beltrami coefficient µ, periodic

with respect to the 2n-adic grid, which satisfies nI

∣∣∣∣2(Sµ)′

ρH(z)

∣∣∣∣2 |dz|2y > Σ2 − C/n (4.6)

on every n-box nI .

The quantitative estimate will be exploited in Section 5.

4.1. Periodic Beltrami coefficients. Given two intervals I, J ⊂ R, let

LI,J(z) = Az +B be the unique linear map with A > 0, B ∈ R that maps I

to J . For a box , we denote its reflection in the real line by . Suppose µ

is a Beltrami coefficient supported on the lower half-plane. We say that µ

is periodic (with respect to the dyadic grid) if for any two dyadic intervals

I, J ⊂ R with |I|, |J | 6 1, µ|I = L∗I,J(µ|J ). We typically assume that µ is

supported on the strip

w : −1 < Imw < 0,

in order for µ to be invariant under translation by 1. In this case, µ descends

to a Beltrami coefficient on the disk via the exponential mapping, which is

16 O. IVRII

eventually-invariant under z → z2. The notion of a Beltrami coefficient

periodic with respect to the p-adic grid is defined similarly.

Before continuing further, we define a dyadic box in the unit disk to be

the image of I , |I| 6 1, under the exponential mapping ξ(w) = exp(2πiw).

Reflecting in the unit circle, we obtain a dyadic box in the exterior unit disk.

Note that these boxes are not geometric rectangles, nor do they tile D or D∗

completely.

4.2. Dynamical Beltrami coefficients. We now consider two classes of

dynamical Beltrami coefficients on the unit disk that naturally arise in com-

plex dynamics and Teichmuller theory:

• MB =⋃f Mf (D) consists of Beltrami coefficients that are eventually-

invariant under some finite Blaschke product

f(z) = z

d−1∏i=1

z − ai1− aiz

,

i.e. Beltrami coefficients which satisfy f∗µ = µ in some open neigh-

bourhood of the unit circle.

• MF =⋃

ΓMΓ(D) consists of Beltrami coefficients that are invariant

under some co-compact Fuchsian group Γ, i.e. γ∗µ = µ for all γ ∈ Γ.

Suppose µ belongs to one of the two classes of Beltrami coefficients above,

with ‖µ‖∞ < 1. We view f = wµ as a conformal map of the exterior unit

disk. From the construction, the image of the unit circle f(S1) is a Julia set

or a limit set of a quasi-Fuchsian group. Using the ergodicity of the geodesic

flow on the unit tangent bundle T1X (Fuchsian case) or Riemann surface

lamination XB (Blaschke case), it is not hard to show that for any ε > 0,

there exists n0 sufficiently large,

σ2(log f ′)− ε < nI

∣∣∣∣2nfρ∗ (z)

∣∣∣∣2 ρ∗|dz|2 < σ2(log f ′) + ε (4.7)

for any n-box nI ⊂ D∗ with n > n0. Applying Theorems 2.1 and 3.2 shows

that Theorem 1.1 holds for conformal maps f = wµ with µ ∈ MB or MF.

More generally, one can prove (4.7) for conformal maps to simply-connected

domains bounded by Jordan repellers, see [1, Section 8] for a definition. The

reader interested in working out the details can consult [22].


5. Applications to integral means

In this section, we use martingale techniques to study the rescaled integral

means spectrum

B∗(τ) = limk→0

Bk(τ/k) = sup|µ|6χH

βS#µ(τ),

near τ = 0. Let B be the dyadic martingale associated to the Bloch function

b = S#µ, |µ| 6 χH and S = ReB be its real part.

Since B∗(τ) is radially invariant, it suffices to estimate βS(t) with t ∈ R.

We view S as a p-adic martingale with p = 2n, where the parameter n will

be chosen momentarily. Suppose I is a 2n-adic interval and I1, I2, . . . , I2n

are its 2n-adic children. Letting ∆j = SIj − SI ,

1

2n

2n∑j=1

exp(t∆j) = 1 +t2

2

(1

2n

2n∑j=1

∆2j

)+∑k>3

tk

k!

(1

2n

2n∑j=1

∆kj

),

= 1 +t2

2Var2n,I S +O

(∑k>3

tk

k!· (Cn)k/2

).

Above, we used (2.9) to estimate the remainder term. Making use of Lemma

4.2(i), we see that the above expression is bounded by

6 1 +nt2

2

(Σ2 log 2

2+O(1/

√n) +O(tn1/2) + . . .

).

Note that in order to use martingale techniques, we had to downgrade the

box estimate with Σ2/2 + C/n to the variance bound Var2n,I S/(n log 2) 6

Σ2/2 + C/√n, cf. Theorem 3.2. Hence,

1

n log 2log

[1

2n

2n∑j=1

exp(t∆j)

]6t2

2

(Σ2

2+O(1/

√n) +O(tn1/2) + . . .

).

Taking n = bt−1c leads to the estimate

B∗(t) 6 Σ2t2/4 +O(|t|5/2). (5.1)

By using Lemma 4.2(ii), one can also use the above reasoning to obtain

a lower bound for integral means spectrum, thus showing that (5.1) is an

equality. As mentioned in the introduction, if one avoids martingales, one

can obtain a remainder term better than O(|t|5/2).

Recently, Hedenmalm [8] proved the beautiful estimate B∗(τ) 6 |τ |2/4for all τ ∈ C. While not sharp near the origin, this estimate cannot be

18 O. IVRII

improved for general values of τ , e.g. B∗(2) = 1 since Bk(2/k) = 1 for all

0 < k < 1.

6. A central limit theorem

Suppose b = log f ′ with f ∈ Sk. Consider the function

br(θ) :=b(reiθ)√| log(1− r)|

. (6.1)

The sub-Gaussian estimate (2.8) shows that most of the integral´

(Re br(θ))2dθ

comes from the set

Aδ =θ : −1/δ < Re br(θ) < 1/δ

,

that is, by making δ > 0 small, we can guarantee thatˆAcδ

(Re br(θ))2dθ < ε,

with the estimate being uniform over all functions b of the form above. In

this section, we show that if σ2(b, r) is close to Σ2(k), then the distribution

of Re br(θ) is close to a Gaussian of mean 0 and variance Σ2(k)/2.

Theorem 6.1. Let 0 < k < 1 be fixed. Suppose f ∈ Sk and b = log f ′.

Given ε > 0, there exists a δ > 0 such that if r is sufficiently close to 1 and

σ2(b, r) =1

2π| log(1− r)|

ˆ|z|=r

|b(z)|2 |dz| > Σ2(k)− δ, (6.2)

then for any t ∈ R, ∣∣P(Re br(θ) < t)−N0,Σ2(k)/2(t)∣∣ < ε. (6.3)

The same statement holds with Im br(θ) as well.

Converting to the upper half-plane, let B be the p-adic martingale associ-

ated to the Bloch function b = log f ′, f ∈ Hk and S = ReB be its real part.

In view of Theorem 3.2 and Lemma 4.1, we can choose p = 2m sufficiently

large to guarantee (p)I

∣∣∣∣2b′ρH (z)

∣∣∣∣2 |dz|2y < Σ2(k) + δ1/2, (6.4)

∣∣∣∣∣ Varp,I S

χ− 1

2

(p)I

∣∣∣∣2b′ρH (z)

∣∣∣∣2 |dz|2y∣∣∣∣∣ < δ1/2, (6.5)


for all p-adic intervals I. Here, χ = log p is the Lyapunov exponent of the

p-adic grid and

(p)I =

w : Rew ∈ I, Imw ∈

[p−1|I|, |I|

]denotes the p-adic box associated to I.

6.1. Characteristic functions. As is standard [6], to prove the central

limit theorem, we need to examine the characteristic functions

ϕn(t) = E exp

(i · tSn√

nχ

), t ∈ R.

Even though martingale jumps are usually not independent, in extremal

situations, one can instead leverage the fact that the local variance is ap-

proximately constant. Observe that if I is a p-adic interval and I1, I2, . . . , Ip

are its children, then for t small,

1

p

p∑j=1

eit∆j = 1− t2

2Varp,I S +O(t3).

The above formula shows that if a martingale S has constant local variance,

that is if Varp,I S/χ = σ2 for all p-adic intervals I, then

ϕn =

[1− σ2t2

2n+O

(t3

n3/2

)]n.

Fixing t ∈ R and taking n→∞ yields

ϕ = limn→∞

ϕn = exp

(−σ

2t2

2

),

which is the characteristic function of the Gaussian N0,σ2 . In this case,

continuity properties of characteristic functions (e.g. [6, Theorem 3.3.6])

imply that the distributions of Sn/√nχ converge to N0,σ2 as n→∞.

More generally, if the local variance is pinched ,

σ2 − δ1 6 Varp,I S/χ 6 σ2 + δ1, ∀I, (6.6)

then

ϕn(t) =

[1− σ2

nt2

2n+O

(t3

n3/2

)]n, with |σn − σ2| 6 δ1. (6.7)

Like in the case of constant local variance, we can still conclude that∣∣P(Sn/√nχ < t)−N0,σ2(t)

∣∣ 6 ε(δ1), for n > n0(p, ε) large,

with ε(δ1)→ 0 as δ1 → 0.

20 O. IVRII

6.2. Allowing bad boxes. For the problem at hand, we must allow a small

proportion of p-adic intervals I to be bad for which the lower bound in (6.6)

fails. Set

n[0,1] =w : Rew ∈ [0, 1], Imw ∈ [p−n, 1]

.

Call a p-adic box (p)I good if the lower bound in (6.6) holds and bad other-

wise. Fix n > 1 and let E ⊂ n[0,1] denote the union of bad boxes contained

in n[0,1].

Lemma 6.2. Suppose S is a real-valued p-adic martingale on [0, 1] with

Varp,I S/χ 6 σ2 + δ1, ∀I, (6.8)

Varp,I S/χ > σ2 − δ1, I good, (6.9) n

[0,1]

χE ·|dz|2

y. δ2, E =

⋃I bad

(p)I . (6.10)

Given any ε > 0, one can choose the parameters δ1 and δ2 small enough to

guarantee∣∣P(Sn/√nχ < t)−N0,σ2(t)

∣∣ 6 ε(δ1, δ2), for n > n0(p, ε) large.

To prove the above lemma, we will use the following elementary identity:

if X is a p-adic martingale on [0, 1], then

σ2(X,n)/χ :=1

nχ

ˆ 1

0(Xn)2 dx =

X2[0,1]

nχ+

n

[0,1]

Varp,I(z)X ·|dz|2

y, (6.11)

where I(z) is the p-adic interval I for which z ∈ (p)I . This identity is a

simple consequence of the orthogonality of increments´ 1

0 ∆〈k1〉X (x)∆

〈k2〉X (x)dx

for k1 6= k2.

Proof. Write S = Sgood +Sbad as a sum of two martingales, where the local

variance of Sgood is close to σ2 on all intervals while the increments of Sbad

are non-zero only on bad intervals. We may form Sgood from S by adjusting

the jumps on the bad intervals, and defining Sbad := S − Sgood to be the

difference. From the construction, it is clear that

σ2(Sbad, n)/χ =1

nχ

ˆ 1

0(Sbadn )2 dx . δ2 · (σ2 + δ1),

which shows that Sbadn /√nχ is small outside of a set of small measure.

Therefore, the distribution of Sn/√nχ is roughly that of Sgood

n /√nχ, which

we already know to be approximately Gaussian.


6.3. Conclusion. Having Lemma 6.2 at our disposal, the proof of Theorem

6.1 is almost immediate. We now resume the discussion from the beginning

of this section. Recall that S = ReB is the p-adic martingale associated to

a Bloch function b = log f ′ with f ∈ Hk, and p was chosen to guarantee

(6.4), (6.5).

Proof of Theorem 6.1. Inspecting (3.7) and making use of (6.4), (6.5) and

(6.11), we see that if the inequality

σ2(S, n)/χ =1

nχ

ˆ 1

0S2n dx+ o(1) > (Σ2(k)− δ)/2 (6.12)

holds for some 0 < δ < δ1, then the exceptional set E defined in the previous

section satisfies n

[0,1]

χE ·|dz|2

y. δ/δ1, with σ2 = Σ2(k)/2. (6.13)

Since δ/δ1 can be made arbitrarily small by requesting δ to be small, Lemma

6.2 is applicable and shows that if n > n0 is sufficiently large, then the

distribution of Sn/√nχ is close to that of the Gaussian N0,Σ2(k)/2. Taking

advantage of the approximation property of Bloch martingales (3.3), we have

proved that the statement

1

| log y|

ˆ 1

0|Re b(x+ iy)|2 dx > (Σ2(k)− δ)/2, y = p−n, (6.14)

implies that Re by(x) := Re b(x + iy)/√| log y| ≈ N0,Σ2(k)/2 in distribution.

For general y, not necessarily of the form p−n, the Lipschitz property of

Bloch functions implies Re by(x) ≈ Re bp−n(x), with n = blogp 1/yc. Finally,

to replace Re b with b in (6.14), we appeal to the complexification relation

(3.9), see also (3.16). This completes the proof.

The proof of Theorem 1.4 is similar except one considers characteristic

functions of two variables

ϕn(s, t) = E exp

(i · sReB + t ImB

√nχ

)and uses the approximate orthogonality (3.10) between ReB and ImB to

show ϕn(s, t) ≈ exp(−Σ2(k)(s2+t2)

4

). We leave the details to the reader.

22 O. IVRII

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Department of Mathematics and Statistics, University of Helsinki, P.O.Box 68, FIN-00014, Helsinki, Finland

E-mail address: [email protected]


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